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A solid floats in a liquid at $20^\circ {\text{C}}$ with $75\% $ of its volume immersed. When the liquid is heated to $100^\circ {\text{C}}$, the same liquid floats with $80\% $ of its volume immersed in the liquid. Calculate the coefficient of expansion of the liquid (Assume the volume of the solid to be constant)
1) $8.33 \times {10^{ - 4}}\;/^\circ {\text{C}}$
2) $83.3 \times {10^{ - 4}}\;/^\circ {\text{C}}$
3) $833 \times {10^{ - 4}}\;/^\circ {\text{C}}$
4) $0.833 \times {10^{ - 4}}\;/^\circ {\text{C}}$

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Last updated date: 05th Mar 2024
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IVSAT 2024
Answer
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Hint: The above problem can be solved by using the concept of thermal expansion. The thermal expansion is the variation in the size of the object due to variation in temperature of the object. The thermal expansion may be linear expansion, surficial expansion and volumetric expansion. The expansion of the object depends on the temperature and coefficient of expansion.

Complete step by step answer
Given: The initial temperature of the liquid is ${T_1} = 20^\circ {\text{C}}$, the final temperature of the liquid is ${T_2} = 100^\circ {\text{C}}$, the immersed volume of the object at initial temperature is ${V_1} = 0.75V$, the immersed volume of the object at final temperature is ${V_2} = 0.8V$.

The formula to calculate the coefficient of expansion of the liquid is given as:
$\gamma = \dfrac{{{V_2} - {V_1}}}{{{V_1}\left( {{T_2} - {T_1}} \right)}}$
Substitute $0.8V$for ${V_2}$ , $0.75V$for ${V_1}$ , $20^\circ {\text{C}}$ for ${T_1}$ and $100^\circ {\text{C}}$ for ${T_2}$ in the above expression to find the coefficient of expansion of the liquid.
$\gamma = \dfrac{{0.8V - 0.75V}}{{\left( {0.75V} \right)\left( {100^\circ {\text{C}} - 20^\circ {\text{C}}} \right)}}$
$\gamma = 8.33 \times {10^{ - 4}}\;/^\circ {\text{C}}$

Thus, the coefficient of expansion of the liquid is $8.33 \times {10^{ - 4}}\;/^\circ {\text{C}}$ and option (a) is the correct answer.

Additional Information The coefficient of expansion of the solid objects may be of three types. First is the coefficient of linear expansion, second is coefficient of surficial expansion and third is coefficient of volumetric expansion.

Note: The coefficient of thermal expansion is the sensitive property, so material should be placed in the isolated space to determine the experimental value of the coefficient of thermal expansion. The coefficient of volumetric expansion is three times of coefficient of linear expansion.